Hello once again everyone. Yep, time for another guide. Talking to the students I tutor (and some friends I did the HSC with), there is one massive time waster in exams. Staring at geometry proofs and wondering, "You want me to prove what?" And you stare for 10 minutes and end up scribbling a proof that isn't really a thing and... It's not pleasant. These remain my pet hate in Mathematics. But there is a lot you can do to prepare yourself and make these proofs a lot easier. Of course, while reading this guide, if anything is unclear,
register and
pop a question below! Also, remember to check out the
notes for both 2U and Extension, which go into more depth than I do if you need that extra little bit of assistance.
Okay, there is lots to cover in this section, and it is difficult to find examples which focus on ONE area. So we'll summarise the content, then do examples, and we'll begin with polygons. You should try and be familiar with the basic properties of the quadrilaterals, including their area formulas, and what you need to have in a proof for that shape (i.e., how do you prove a shape is a rectangle?). Here's a brief summary:
You should notice what you need to prove for the various shapes, lots of people waste time by proving too much!
You should know the formula for finding the angle sum of polygons, and
remember that all the angles of a regular polygon (all sides equal) are equal! \quad \\ \\ \theta=\frac{S_\theta}{n})
Polygons aren't covered as much in the HSC, but they are always in the exam in some form!
Then we come to triangles. Now this guide won't cover trigonometry, and geometry questions normally won't overlap this area either. However, keep in mind the trig ratios may be necessary. The only thing you'll need here is a knowledge of the shape's properties:
- Angle sum is 180 degrees
- Exterior angle of a triangle is equal to the sum of the interior angles
And actually, that's most of the work done. Usually, you will be proving congruence (the same triangle, same size) or similarity (the same triangle, but possibly of different size). The proofs for congruence are, we remember,
RHS, AAS, SAS, SSS. For similarity, we need to prove either that they are equiangular, or that the sides are all in proportion. Similarity is much more common to prove.
Finally, we have parallel line related proofs. Make sure you remember the following rules:
- Corresponding angles are equal
- Alternate angles are equal
- Co-Interior angles add to 180
- Sets of parallel lines cut transversals in proportion
Right! That was a massive amount of content, and there is so many weird tidbits that I very well may have forgotten something. There is also lots of little tricks to these properties, so if you have any useful ones,
share them below! Let's work through a tricky example to demonstrate some of this.
Example One (2014 HSC): In Triangle DEF, a point S is chosen on the side DE. The length of DS is x, and the length of ES is y. The line through S parallel to DF meets EF at Q. The line through S parallel to EF meets DF at R. The area of triangle DEF is A. The areas of triangle DSR and triangle SEQ are \(A_1\) and \(A_2\) respectively. Thank goodness for diagrams right?
a) Show that Triangle DEF is similar to Triangle DSR. Okay, this is a good chance to mention something important:
make your reasoning clear! To emphasise this, I'm going to do this proof in words (you would use symbols):
Angle D is common to both triangles
Angle DFE is equal to Angle DRS (since they are equal corresponding angles on the parallel lines RS//FE)
Since two angles in the triangles are equal, the third is also, since both triangles have an angle sum of 180
Therefore the triangles are equiangular, and thus, similar
Everything that was there should be in a proof to guarantee full marks, just with symbols instead of words where appropriate.
b) Prove that \(\frac{DR}{DF}=\frac{x}{x+y}\)Geometry questions demonstrate more than any others that it is absolutely essential to pay attention to the previous parts of a question. We have proved similarity, so we know that the sides of the triangles are in proportion. State this, rewrite out the answer, and you have full marks
c) Prove that \(\sqrt{\frac{A_1}{A}}=\frac{x}{x+y}\) We consider the areas of Triangle DRS and Triangle DFE in terms of the formula \(A=\frac { 1 }{ 2 } ab\sin { C }\). Remember this one too!
Using the previous part, we can get the answer:
^2\\ \sqrt{\frac{A_1}{A}}=\quad \frac{x}{x+y} )
For full marks, this proof would require mention that the ratio DS/DE is equal to the ration DR/DF (since the triangles are similar), and that in the last step, we take the positive square root since the areas are positive.
I have omitted the last part of this question, which is more an algebra trick, but we can see how various areas of above are combined to contribute to a fairly complex proof.
Another common question is bearings.
Example 2: Chris leaves island A in a boat and sails 142 km on a bearing of 078° to island B. Chris then sails on a bearing of 191° for 220 km to island C, as shown in the diagram. 
a) Show that the distance from island C to island A is approximately 210 km. Right, this time I'll do it exactly as you should do it in the HSC/Trials (all angles are in degrees):
}\\\therefore\quad\angle ABC=67 \text{(angles at point P add to 360)})
Therefore, by the cosine rule, the required distance d is:
This question shows that the sine and cosine rule come up a lot in bearings questions. A reminder of those rules:
The final piece of the puzzle is basic circle geometry. These are formulaic questions, and 2U students need only remember that radii at all points of a circle are equal, and the following formulae:
)
Most questions involve combining these formulas (EG- relating arc length and area is a big one). Be careful with the algebra and these questions should be a snap. And my biggest tip,
remember to reset your calculator to radians for this section! Extension students, there is a short guide for all of your circle geometry stuff on its way, stay tuned
Geometry has the potential to be the hardest part of your exam, so to close this guide, I want to stress my
biggest tip for dealing with geometry proofs that you don't get straight away. Leave them until the end. This way they will stew in the back of your mind, and when you come back to them, you'll know everything else is done, and you aren't feeling as pressured. Trust me, it works wonders. Thanks for reading guys! Be sure to ask questions, check out the notes, and study study study!
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